The real-quadratic Rabinowitsch problem at the two-prime-divisor level: determine all $d\equiv 2\pmod 4$ for which $d-x^2$ has at most two distinct prime divisors for every admissible even $x\le\sqrt d$

The classical Frobenius–Rabinowitsch theorem links prime-producing quadratic polynomials to class number one for imaginary quadratic fields, and its real quadratic counterpart — polynomials $(d-x^2)/4$ producing primes for $d\equiv 5\pmod 8$ — was completely classified through work of Biró, Byeon–Kim–Lee, and Biró–Granville. The open successor problem, posed for even $d$ by Louboutin (arXiv:2511.15243, 2025) following Gica's relaxation for imaginary fields, weakens primality to having at most two distinct prime divisors: determine all $d\equiv 2\pmod 4$ such that $\omega(d-x^2)\le 2$ for every even $2\le x\le\sqrt d$. For square-free $d\equiv 2\pmod 4$ the property forces the class number of $\mathbb{Q}(\sqrt d)$ to divide 16, yet even finiteness of the set of such $d$ is unproved, because the property is not known to bound the fundamental unit; the conjectured list of 16 values is open.

Background

For a square-free integer $d\ne 1$, the quadratic field $K=\mathbb{Q}(\sqrt d)$ has ring of integers $\mathbb{Z}[\sqrt d]$ ($d\equiv 2,3\pmod 4$) or $\mathbb{Z}[(1+\sqrt d)/2]$ ($d\equiv 1\pmod 4$). Its ideal class group $\mathrm{Cl}(K)$ — the group of fractional ideals modulo principal ideals — is finite; its order $h(K)$ is the class number, and $h(K)=1$ exactly when every ideal is principal, i.e. unique factorization holds at the level of ideals. The narrow class group refines this by requiring principal ideals to be generated by an element of positive norm; the narrow class number $h^+(K)$ equals $h(K)$ precisely when $K$ contains a unit of norm $-1$. Determining all fields with class number one is Gauss's class number problem: for imaginary quadratic fields ($d<0$) it was solved by Heegner, Baker, and Stark (see Goldfeld's survey), but for real quadratic fields it remains open, with infinitely many expected.

Rabinowitsch (1913) discovered an arithmetic encoding of the imaginary case: for square-free $d\equiv 3\pmod 4$, the norm-form values $(d+x^2)/4=N((x+\sqrt{-d})/2)$ are prime or 1 for all odd $1\le x\le\sqrt d$ if and only if $d$ is prime and $h(\mathbb{Q}(\sqrt{-d}))=1$ — the Frobenius–Rabinowitsch criterion, explaining Euler's prime-rich polynomial $n^2+n+41$ via the field $\mathbb{Q}(\sqrt{-163})$. The complete list of such $d$ follows from the Heegner–Baker–Stark solution of the imaginary class number one problem (history in Goldfeld's survey; class-number tables in Watkins 2004): $d\in\{3,7,11,19,43,67,163\}$.

For real quadratic fields the honest analogue, proved by Louboutin (1990, Theorem 2, as quoted in arXiv:2511.15243), is more constrained: for $0<d\equiv 5\pmod 8$, the values $(d-x^2)/4$ are prime or 1 for all odd $3\le x\le\sqrt d$ if and only if $d$ is square-free of Richaud–Degert type — $d=m^2+4$, $m^2-4$, or $4m^2+1$, the families of real quadratic fields whose fundamental unit $\varepsilon_d$ (the smallest unit $>1$, generator of the unit group up to signs) is small, $\log\varepsilon_d\ll\log d$ — and $h(\mathbb{Q}(\sqrt d))=1$. Prime production therefore forces class number one (one-direction criteria of this kind go back to Mollin–Williams and were extended by Byeon–Stark and Ramírez), but class number one alone is not sufficient (e.g. $d=61$: $h=1$ but $(61-25)/4=9$), and the correct hypothesis is the ordinary, not narrow, class number. By the Brauer–Siegel theorem (which links $h(K)\log\varepsilon_d$ to $\sqrt d$), small units make $h=1$ possible for only finitely many $d$, and this finiteness route was completed: Biró's proofs of Yokoi's and Chowla's conjectures and Byeon–Kim–Lee's proof of Mollin's conjecture — three conjectures, each asserting that within one of the Richaud–Degtert families class number one occurs only for an explicitly listable set of parameters — together with Biró–Granville's evaluation of the zeta functions attached to the ideal classes of real quadratic fields at $s=0$, prove the Rabinowitsch list for real fields is exactly $d\in\{13,21,29,37,53,77,101,173,197,293,437,677\}$ — a complete classification.

The live frontier weakens primality. Write $\omega(n)$ for the number of distinct prime divisors of $n$ (with $\omega(1)=0$). On the imaginary side, Gica (2024) proposed replacing primality of the norm-form values by the condition $\omega(d+x^2)\le 2$ (at most two distinct prime divisors) for all $x\le\sqrt d$ of the right parity, and determined the qualifying $d$ with some gaps; Louboutin (2025) corrected the gaps, settled the cases of odd $d$ (Theorems 1–3; the case $d\equiv 7\pmod 8$ unconditionally except for at most one exceptional value, by the Siegel–Tatuzawa theorem, with no exception under the Riemann hypothesis for the relevant Dedekind zeta functions) and of $d\equiv 2\pmod 4$ with even $x$ (Theorems 4–6), using the known determination of the imaginary quadratic class groups of order dividing 16, and left the odd-$x$ case for $d\equiv 2\pmod 4$ as his open Conjectures 7–8. He then began the real-field transposition with $f_d(x)=d-x^2=-N(x+\sqrt d)$. Note that for $d\equiv 2\pmod 4$ and even $x$ one has $d-x^2=2m$ with $m$ odd, so $\omega(d-x^2)\le 2$ equivalently requires $(d-x^2)/2$ to be 1 or a prime power — a relaxation of the 'prime or 1' condition in the imaginary $d\equiv 2\pmod 4$ criterion recalled in the same introduction ('$(d+x^2)/2$ prime or 1 for all even $0\le x\le\sqrt d$ if and only if $d/2$ is prime and the class number of $\mathbb{Q}(\sqrt{-d})$ equals two, giving $d\in\{2,6,10,22,58\}$'). Setting $M'_{\mathrm{even}}(d)=\max\{\omega(d-x^2): 2\le x\le\sqrt d,\ x\text{ even}\}$, he proved (Theorem 9) that square-free $d\equiv 2\pmod 4$ with $M'_{\mathrm{even}}(d)\le 2$ satisfy $h(\mathbb{Q}(\sqrt d))\mid 16$, and (Theorem 11) that the non-square-free such $d$ are exactly $\{18,50,54,90,98\}$. But for square-free $d$ he could not prove that the property forces a small fundamental unit, so Brauer–Siegel cannot even establish finiteness, and the completeness of the experimentally found list is conjectural (Conjecture 10). A related odd-$d$ direction lies outside the present problem: for $d\equiv 5\pmod 8$ and odd $x$ one has $d-x^2\equiv 4\pmod 8$, so the raw condition $\omega(d-x^2)\le 2$ is equivalent to $(d-x^2)/4$ being 1 or a prime power — strictly weaker than the solved classical primality condition (e.g. $d=61$ satisfies the raw condition while $(61-25)/4=9$) — and no classification of this weaker condition exists in the literature; Louboutin's real-field program (Theorem 9, Conjecture 10, Theorem 11) is confined to the even scope $d\equiv 2\pmod 4$ with even admissible $x$, and the present problem retains that scope.

Problem Statement

Let $\omega(n)$ denote the number of distinct prime divisors of an integer $n\ge 2$, and set $\omega(1)=0$. The problem is posed for even $d\equiv 2\pmod 4$, where an admissible $x$ is an even integer with $2\le x\le\sqrt d$ (Louboutin's $M'_{\mathrm{even}}(d)$). Here $d$ is a positive integer that is not a perfect square (automatic, since squares are congruent to $0$ or $1$ modulo $4$), so $d-x^2\ge 1$ for every admissible $x$. Determine completely the set of $d\equiv 2\pmod 4$ such that $\omega(d-x^2)\le 2$ for every admissible $x\le\sqrt d$. The problem, posed in arXiv:2511.15243 (2025) as the real quadratic analogue of Gica's imaginary-field classification, has one scientific objective: a Frobenius–Rabinowitsch-type classification of the real quadratic fields $\mathbb{Q}(\sqrt d)$ at the two-prime-divisor level. The non-square-free case is already settled (Louboutin's Theorem 11: the qualifying non-square-free $d\equiv 2\pmod 4$ are exactly $\{18,50,54,90,98\}$); the open core is the square-free case $d\equiv 2\pmod 4$: prove or refute that the set of such $d$ equals Louboutin's Conjecture-10 list $\{2,6,10,14,22,26,30,38,42,62,110,122,182,278,362,398\}$, or prove that the set is infinite if such is the truth. Finiteness itself is currently out of reach, because $\omega(d-x^2)\le 2$ is not known to force a small fundamental unit in $\mathbb{Q}(\sqrt d)$, blocking the Brauer–Siegel argument that worked for the Richaud–Degert families.

An answer must determine the set exactly (or prove it infinite, if such is the truth), not merely extend the experimental list.

References

1. G. Rabinowitsch, 'Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen Zahlkörpern', Journal für die reine und angewandte Mathematik 142 (1913), 153–164. DOI: 10.1515/crll.1913.142.153, https://doi.org/10.1515/crll.1913.142.153
2. D. Goldfeld, 'Gauss's class number problem for imaginary quadratic fields', Bulletin of the American Mathematical Society (N.S.) 13 (1985), 23–37. DOI: 10.1090/S0273-0979-1985-15352-2, https://doi.org/10.1090/S0273-0979-1985-15352-2
3. S. Louboutin, 'Prime producing quadratic polynomials and class-numbers of real quadratic fields', Canadian Journal of Mathematics 42 (1990), 315–341. DOI: 10.4153/CJM-1990-018-3, https://doi.org/10.4153/CJM-1990-018-3
4. D. Byeon and H. M. Stark, 'On the finiteness of certain Rabinowitsch polynomials', Journal of Number Theory 94 (2002), 177–180. DOI: 10.1006/jnth.2001.2729, https://doi.org/10.1006/jnth.2001.2729
5. A. Biró, 'Yokoi's conjecture', Acta Arithmetica 106 (2003), 85–104. DOI: 10.4064/aa106-1-6, https://doi.org/10.4064/aa106-1-6
6. A. Biró, 'Chowla's conjecture', Acta Arithmetica 107 (2003), 179–194. DOI: 10.4064/aa107-2-5, https://doi.org/10.4064/aa107-2-5
7. M. Watkins, 'Class numbers of imaginary quadratic fields', Mathematics of Computation 73 (2004), 907–938. DOI: 10.1090/S0025-5718-03-01517-5, https://doi.org/10.1090/S0025-5718-03-01517-5
8. D. Byeon, M. Kim and J. Lee, 'Mollin's conjecture', Acta Arithmetica 126 (2007), 99–114. DOI: 10.4064/aa126-2-1, https://doi.org/10.4064/aa126-2-1
9. A. Biró and A. Granville, 'Zeta functions for ideal classes in real quadratic fields, at s = 0', Journal of Number Theory 132 (2012), 1807–1829. DOI: 10.1016/j.jnt.2012.02.003, https://doi.org/10.1016/j.jnt.2012.02.003
10. V. J. Ramírez, 'Prime producing quadratic polynomials associated with real quadratic fields of class-number one', Periodica Mathematica Hungarica 86 (2022), 495–502. DOI: 10.1007/s10998-022-00487-1, https://doi.org/10.1007/s10998-022-00487-1
11. A. Gica, 'Class numbers, Ono invariants and some interesting primes', Indagationes Mathematicae 35 (2024), 1249–1258. DOI: 10.1016/j.indag.2024.06.003, https://doi.org/10.1016/j.indag.2024.06.003 (arXiv:2401.10930, https://arxiv.org/abs/2401.10930)
12. S. R. Louboutin, 'Ideal class groups of some quadratic number fields and factorization of values of some quadratic polynomials', preprint arXiv:2511.15243 (2025). arXiv: 2511.15243, https://arxiv.org/abs/2511.15243